24V proximitiy switch, NPN transistor, Arduino interrupt input

Use 10kΩ for R3.

I didn't look up a BC547, but if that's a typical small signal transistor as it should be, figure a gain of at least 50. 6V supply minus 700mV B-E drip leave 5.3V accross the resistor. 5.3V / 10kΩ = 530µA. Times the gain of 50, and that can support over 25mA of collector current. That's well more than any reasonable digital input will require, even if it has a pullup connected.

As others have said, you may need a pullup if whatever you are driving doesn't passively make the line go high. However, make sure the pullup goes to same voltage as the power supply of whatever will receive the digital signal. Your supply is 6V, which is probably higher than the receiver's supply. That could cause the receiver to not work or even to damage it if its power voltage is significantly lower.

The circuit you describe is an emitter follower - the emitter voltage follows the base voltage and is always about 0.7 volts negative of the base. The transistor doesn't care where you think "ground" is, its operation only depends on the voltages between its pins.

If you ground the emitter, and put your load between the collector and the positive supply, you will be able to get very close to the supply voltage across the load when the Arduino output is high. You should have a resistor of 1K or so between the Arduino output pin and the transitor base, to limit base current and loading of the Arduino output pin.